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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2013 • Question 3

Question

Math 6420 • PDE Spring 2013 • Question 3

  1. Find all conditions on \(\alpha ,\beta ,\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \tag {1} \] exist.
  2. Find an integral representation for the solution in the case that \(\alpha =0\).

Raw LaTeX

\begin{enumerate}
\item[(a)] Find all conditions on \(\alpha,\beta,\gamma\), and \(f(x)\) for which solutions of
\[
  u''+\alpha^2u=f(x),\qquad u(0)=\beta,\qquad \alpha u'(0)-u'(1)=\gamma
  \tag{1}
\]
exist.

\item[(b)] Find an integral representation for the solution in the case that \(\alpha=0\).
\end{enumerate}

Topic Tags

Cauchy Integral Formula and Integral Theorems Distributions and Weak Solutions Entropy Solutions, Shocks, and Rarefactions Linear Programming and Optimization